Melt Cup together with handle
You could use the bridge function of the loop tools add-on if you have it enabled. You can find it under user  preferences> add-ons if you don't. In the attached gif I used a curve for the handle of my cup, and a cylinder for the cup itself. Press  Alt+C  to convert the curve object to a mesh object, then  Shift and RMB to select both objects. Then press  Ctrl+J to join your meshes. Now you can delete the unnecessary faces, and bridge the appropriate edges. To do so, use  Alt+RMB  to select your first edge loop, then Shift+Alt+RMB to select your second edge loop, and click bridge under the loop tools options.